lie algebras - sl(2)-modules...

You can't assume the module is irreducible, since many aren't!

However, if you want to learn something about modules of $sl_2$ it helps to make the following observations:

1. Each module is a sum of irreducible modules


2. It's easy to describe all irreducible modules.

In particular, when you'll be solving 2, you'll notice that there is exactly one irreducible modle of dimension 2. In fact, there is one of dimension 3, one of dimension 4, etc.